In the context of most discussions on this site, "confidence" is the probability that a guess is correct. For example:

• I guess that a flipped coin will land heads. My confidence is 1/2, because I have arbitrarily picked 1 out of 2 possible outcomes.
• I guess that, when a coin is flipped repeatedly, the ratio of heads will be close to half. My confidence is close to 1, because I know from experience that most coins are fair (and the law of large numbers).

"Confidence interval" is just confidence that something is within a certain range.

You should also be aware that in the context of frequentism (most scientific papers), these terms have different and somewhat confusing technical definitions.